At the cusp bifurcation point two branches of
saddle-node bifurcation curve meet tangentially, forming a semicubic parabola.
For nearby parameter values, the system can have
three equilibria which collide and disappear pairwise via the
saddle-node bifurcations. The cusp bifurcation
implies the presence of a hysteresis phenomenon.

This bifurcation is
characterized by two bifurcation conditions \( \lambda_{1}=0\) and
\(a(0) = 0\) (has codimension two)
and appears generically in two-parameter families of smooth ODEs.
Generically, the critical equilibrium \( x^0 \) is a triple root of the equation
\( f(x,0)=0 \) and \(\alpha=0\) is the origin
in the parameter plane of two branches of saddle-node bifurcation curve.
Crossing each branch results in a pairwise collision and disappearance of equilibria.
These bifurcations are nondegenerate and no more than three equilibria exist in a
neighbourhood of \( x^0 \ .\)

The local bifurcation diagram of the normal form with \(\sigma=-1\) is presented in Figure 2. The point \( \beta=0 \) is the origin of two branches of the
saddle-node bifurcation curve:
\[
LP_{1,2}=\{(\beta_1,\beta_2): \beta_1=\mp \frac{2}{3\sqrt{3}} \beta_2^{3/2},\ \beta_2 > 0 \},
\]
which divide the parameter plane into two regions. Inside the wedge between \( LP_1 \)
and \( LP_2 \ ,\) there are three equilibria, two stable and one unstable. Outside
the wedge, there is a single equilibrium, which is stable. If we approach the cusp point
from inside the wedge, all three equilibria merge together.

The equilibrium manifold of the normal form
\[
{\mathcal M}=\{(y,\beta) \in {\mathbb R}^3: \beta_1 + \beta_2 y - y^3 =0 \}
\]
is shown in Figure 1. The projection of this manifold onto the parameter
plane has fold singularities along \( LP_{1,2} \ ,\) while the cusp singularity
shows up at the origin. Here we have hysteresis:
A jump to a different stable equilibrium happens at either \( LP_{1} \) or
\( LP_{2} \ ,\) depending on whether the traced under variation of \( \beta_1 \)
equilibrium belongs initially to the upper or lower sheet of \( {\mathcal M} \ .\)

The case \( \sigma=1 \) can be reduced to the one above by the substitution
\( t \to -t,\ \beta \to -\beta \)

Multidimensional Case

Figure 3: Cusp bifurcation on the plane in the system\[\dot{y}=\beta_1+\beta_2y -y^3\] and \(\dot{z}=-z\ .\)

In the \(n\)-dimensional case with \(n \geq 2\ ,\) the Jacobian
matrix \(A_0\) at the cusp bifurcation has

a simple zero eigenvalue \(\lambda_{1}=0\ ,\) as well as

\(n_s\) eigenvalues with \({\rm Re}\ \lambda_j < 0\ ,\) and

\(n_u\) eigenvalues with \({\rm Re}\ \lambda_j > 0\ ,\)

with \(n_s+n_u+1=n\ .\)
According to the Center Manifold Theorem, there is a family of smooth
one-dimensional invariant manifolds \(W^c_{\alpha}\) near the origin.
The \(n\)-dimensional system restricted on \(W^c_{\alpha}\) is
one-dimensional, hence has the normal form above.